Communication complexity of set-disjointness for all probabilities

We study SET-DISJOINTNESS in a generalized model of randomized two-party communication where the probability of acceptance must be at least α(n) on yes-inputs and at most β(n) on no-inputs, for some functions α(n) \u3e β(n). Our main result is a complete characterization of the private-coin communication complexity of SET-DISJOINTNESS for all functions a and b, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where α =1=2+ε(n) and β =1=2-ε(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a “restricted” class SBP (which lies between MA and AM) and an “unrestricted” class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of SET-DISJOINTNESS